In Chapter 2 we wrote a system of It near equal ions using matrix notation. Similarly, we can use matrix notation to write a system of linear inequalities.
(a) Find matrices A, B, C, and X such that the maximization problem in Example 1 of Section 4.1 can be written as
Maximize CX
subject to: AX ≤ B
with X ≥ O.
(Hint: Let B and X he column matrices, and C a row matrix.)
(b) Show that the dual of the problem in part (a) can be written as
Minimize YB
subject to: YA ≥ C
with Y ≥ O,
where Y is a row matrix.
(c) Show that for any feasible solutions X and Y to the original and dual problems, respectively, CX ≤ YB. (Hint: Multiply both sides of AX ≤ B by Y on the left. Then substitute for YA.)
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